System and method for efficient basis conversion

ABSTRACT

This invention describes a method for evaluating a polynomial in an extension field F q   M , wherein the method comprises the steps of partitioning the polynomial into a plurality of parts, each part is comprised of smaller polynomials using a q −th  power operation in a field of characteristic q; and computing for each part components of q −th  powers from components of smaller powers. A further embodiment of the invention provides for a method of converting a field element represented in terms of a first basis to its representation in a second basis, comprising the steps of partitioning a polynomial, being a polynomial in the second basis, into a plurality of parts, wherein each part is comprised of smaller polynomials using a q −th  power operation in a field of characteristic q; evaluating the polynomial at a root thereof by computing for each part components of q −th  powers from components of smaller powers; and evaluating the field element at the root of the polynomial.

This application is a divisional of U.S. patent application Ser. No.09/948,793 filed on Sep. 10, 2001, which is a continuation ofPCT/CA00/00256 filed on Mar. 13, 2000 and which claims priority fromCanadian Patent Application No. 2,265,389 filed on Mar. 12, 1999, thecontents of which are hereby incorporated by reference.

FIELD OF THE INVENTION

This invention relates to the field of cryptographic systems, andconversion of elements between bases used in such systems.

BACKGROUND OF THE INVENTION

It is well known that there is only one finite field of any given order,but that there are many different representations. When an extensionfield is built by adjoining a root of an irreducible polynomial to theground field, the choice of irreducible affects the representation ofthe extension field. In general if F_(q) _(m) is the finite field, whereq is a prime and F_(q) is the ground field over which it is defined, theelements of the finite field can be represented in a number of waysdepending on the choice of basis. In order to interoperate,cryptographic systems employing finite fields often need to establish acommon representation. In addition to the choice of irreduciblepolynomial, finite fields can also be represented by either polynomialor normal basis. A polynomial basis represents elements of F_(q) _(m) aslinear combinations of the powers of a generator element x: {x⁰, x¹, . .. x^(m−1)}. A normal basis representation represents elements as linearcombination of successive q-th powers of the generator element x: {x^(q)⁵ , x^(q) ¹ , . . . x^(q) ^(m−1) }. Each basis has its own advantages,and cryptographic implementations may prefer one or the other, or indeedspecific types of irreducible polynomials, such as trinomials orpentanomials.

To support secure communication between devices using differentrepresentations, basis conversion, which changes the representation usedby one party into that used by another party is generally required.

Basis conversion often entails the evaluation of a polynomial expressionat a given finite field element. If an element a, when represented as apolynomial, is given as a(x)=Σa_(i)x^(i) mod f(x), where f(x) is anirreducible, in one basis, then the conversion of the element a into anew representation using another irreducible polynomial requires that abe evaluated at r, where r is a root of the new irreducible polynomialin the field represented by f(x), then a(r) is the element a in the newrepresentation. Any of the conjugates of r (the other roots of the newirreducible) will also induce equivalent, but different representations.

There is a need for an efficient method for evaluating thesepolynomials, for application to basis conversion.

SUMMARY OF THE INVENTION

In accordance with this invention there is provided a method forevaluating polynomials in an extension field comprising the steps of:partitioning the polynomials into a plurality of parts, such that eachpart may be computed from smaller polynomials using a q-th poweroperation in a field of characteristic q.

In accordance with a further embodiment of the invention there isprovided a method for evaluating a polynomial in an extension fieldcomprising the steps of computing components of the q-th powers fromcomponents of smaller powers.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features of the preferred embodiments of the inventionwill become more apparent in the following detailed description in whichreference is made to the appended drawings wherein:

FIG. 1 is a schematic diagram illustrating an embodiment of the presentinvention;

FIGS. 2(a) and 2(b) are schematic diagrams illustrating an embodiment ofthe invention;

FIGS. 3(a) and (b) are schematic diagrams of further embodiments of theinvention; and

FIG. 4 is a schematic diagram of a three level tree according to anembodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In a first embodiment of the invention, we exemplify a specific case F₂_(m) of the general field F_(q) _(m) , then in order to evaluate a fieldelement a(x)=Σa_(i)x^(i) in F₂ _(m) , it is noted that approximately onehalf of the exponents x^(i) are even powers. These powers can beefficiently calculated from lower degree exponents of odd powers. Forexample, the powers for i=2, 4, 6, 8, 10 can be calculated by squaringthe powers for i=1, 2, 3, 4, 5, respectively. This approach does notapply to the odd powers, but if we partition a into even and odd powers,e.g. a(x)=a_(even)(x)+a_(odd)(x), and if we factor x from the oddpowers, then a will be represented by a sum of even powers and x times asum of even powers. Explicitly, $\begin{matrix}{{a(x)} = {\left( {a_{0} + {a_{2}x^{2}} + {a_{4}x^{4}} + \ldots} \right) + {x\left( {a_{1} + {a_{3}x^{2}} + {a_{5}x^{4}} + \ldots}\quad \right)}}} \\{= {{a_{even}(x)} + {{xa}_{even}^{\prime}(x)}}}\end{matrix}$where a_(even)′(x) is the even-powered polynomial derived by factoring xfrom add (x).

In a field of characteristic 2, F₂ _(m) squaring is a linear operation,which allows a_(even) and a_(even)′ to be expressed as squares ofpolynomials of smaller (roughly one half) degree. Explicitly, definingb(x)=a₀+a₁x+a₄x+a₆x+ . . . and c(x)=a₁+a₃x+a₅x+a₇x+ . . . , then a(x)can be expressed as a(x)=(b(x))²+x(c(x))². Now b and c haveapproximately half the degree of the original polynomial a to beevaluated.

Evaluation of b (and c) can (recursively) be made by further dividingthe polynomial into odd and even parts. The odd part can be shifted toan even power by factoring out x and expressing the result as acombination of squares of lower degree polynomials. At each applicationof the partitioning and shifting a two squaring operations and onemultiplication by x are required.

At some point, the recursive partitioning can be halted and thecomponent polynomials evaluated via one or more methods.

Note that although we have described the method for the extension fieldsover F₂, similar application can be made to other characteristics if thepolynomials are divided into more components. For example, for theextension held over F₃, the polynomial a(x) might be divided intoa(x)=a_(0 mod 3)+a_(1 mod 3)+a_(2 mod 3), wherea _(0 mod)=(a ₀ +a ₃ x ³ +a ₆ x ⁶ . . . ) a_(1 mod 3)=(a ₁ x+a ₄ x ⁴ +a₇ x ⁷ . . . ), and a_(2 mod 3)=(a ₂ x+a ₅ x ⁵ +a ₈ x ₈ . . . ).In general, for extension fields over F₃, the polynomial may be dividedinto q parts. Even over characteristic 2, the polynomial a might bebroken up into more than 2 polynomials, at the cost of moremultiplications by x or powers of x.

As an example of a preferred embodiment at a given size, considerconversion of polynomial representations over F₂ ₁₆₃ . An element ofthis field is represented by the polynomiala(x)=a ₀ +a ₁ x+a ₂ x ²+ . . . a₁₆₂ x ¹⁶².The first branching divides a(x) into: $\begin{matrix}{{a(x)} = {\left( {a_{0} + {a_{2}x} + {a_{4}x^{2}} + {a_{6}x^{3}} + \ldots + {a_{162}x^{81}}} \right)^{2} +}} \\{{x\left( {a_{1} + {a_{3}x} + {a_{5}x^{2}} + {a_{7}x^{3}} + \ldots + {a_{161}x^{80}}} \right)}^{2}} \\{{= {\left( {b(x)} \right)^{2} + {x\left( {c(x)} \right)}^{2}}},}\end{matrix}$where b(x) and c(x) are each polynomials referred to as componentpolynomials C_(i), C_(j).At the next level down, $\begin{matrix}{{b(x)} = {\left( {b_{0} + {b_{2}x} + {b_{4}x^{2}} + {b_{6}x^{3}} + \ldots + {b_{80}x^{40}}} \right)^{2} +}} \\{{x\left( {b_{1} + {b_{3}x} + {b_{5}x^{2}} + {b_{7}x^{3}} + \ldots + {b_{81}x^{40}}} \right)}^{2}} \\{= {\left( {d(x)} \right)^{2} + {{x\left( {e(x)} \right)}^{2}.}}}\end{matrix}$The polynomial c(x) is partitioned in a similar way.As mentioned above, the partitioning of polynomials into parts may behalted and the resulting component polynomials evaluated at a root byusing any one of several known methods. The positions at which thepartitioning stops may be denoted as the leaves of the evaluation tree.The component polynomials at the leaves may be evaluated eitherdirectly, or with Horner's rule. However, current methods do not exploita property of finite fields of characteristic q, wherein taking the q-thpowers is usually an efficient operation.

An exemplary method will now be described for efficiently evaluating acomponent polynomial for extension fields over F₂.

At the leaf, a component polynomial a(x)=Σa_(i)x^(i) must be evaluatedat a root of a new irreducible. Again, approximately one half of theseterms will be even and can be calculated efficiently from odd powers.These odd powers will either be stored, for explicit basis conversion,or calculated explicitly by multiplication. If, for example a(x) isguaranteed to be of degree not greater than 10 (which may be the case ifcertain evaluation tree depths are employed), then a(x) can be evaluatedfrom the powers 1, 3, 5, 7, 9, which are either stored or calculated.Squarings can be reduced by grouping coefficients together. This isshown schematically in FIG. 2(a) where a notional table is constructedto show the relationship between the stored or evaluated odd powers of rand the higher degree even powers of r. Thus, consider the first row inwhich r², r⁴, and r⁸ are derived by squaring r¹, similarly, r⁶ isderived by squaring r³ and r¹⁰ is derived by squaring r⁵. It is to benoted that in this example, powers of 2 are used.

Turning back to FIG. 2(a), however, the notional table may be used asshown schematically in FIG. 2(b). Thus, assume an accumulator is setinitially to 0. Since we are using an extension field over F₂ thecoefficients a_(g) are either 0 or 1. First, if a_(d) is 1, then r¹ isadded to the accumulator, which consists of a copying operation in aprocessor. Next, the accumulator is squared. Next, if a₄ is 1, then r¹is added into the accumulator. Again, the accumulator is squared. Now,if a₂, a₆, a₁₀ are one (1) then r¹, r³, r⁵ are added into theaccumulator respectively. Again, the accumulator is squared. Finally, ifa₀, a₁, a₃, a₅, a₇, a₉ are set (1), then r⁰, r¹, r³, r⁵, r⁷, r⁹ areadded into the accumulator. This completes the evaluation of a(x) at r,requiring three squares and the initial evaluation of r⁰, r¹, r³, r⁵,r⁷, r⁹, which can be reused at another leaf evaluation.

It will be apparent to those skilled in the art that the precomputedvalues technique can equally well be applied to multiplicationtechniques.

For polynomials of larger degrees, similar evaluations can be made fromevaluation of odd powers. First, the coefficients of those exponentswith the largest powers of 2 are added into the accumulator according towhether the polynomial to be evaluated has non-zero coefficients atthose powers, then the result is squared. Next, powers divisible by oneless power of 2 are added in as required by polynomial evaluation.Accumulation and squaring continues until the odd powers themselves areadded in as required by the polynomial under evaluation.

In FIGS. 3(a) and 3(b), a similar evaluation is exemplified for anextension field over F₃ and for a polynomial of degree no greater than17. Note that in this embodiment, the coefficients a₁ may take a value0, 1, or 2. Thus, the powers are added with the required coefficients.In general then, for an extension field over F_(q), powers of q are usedto construct the notional table and evaluation of the polynomialproceeds by accumulation and q powering until all required powers in thepolynomial are added in as required by the polynomial being evaluated.

An application of the above method to basis conversion may beillustrated as below. Given a field F₂ ₃₁ and a pair of bases havingrespective irreducible f₁ and f₂ and if f₁=x³¹+r⁶+1; and f₂=x³¹+x³+1.Then, a root of f₁ in the field represented by f₂ is given byr=x²⁶+x²⁴+x²³+x²²+x¹⁹+x¹⁷+x¹²+x¹¹+x⁹+x³+x⁶+x⁵+x³+x². Now, to convert anelement a(x)=a_(f) ₁ in the first basis to a representation in thesecond basis a_(f) ₂ (that is to basis defined by f₂) we proceed asfollows. Let ${a(x)} = {\sum\limits_{i = 0}^{30}{a_{i}x^{i}}}$in general. For this example, we choose a specific element:a(x)=x ³⁰ +x ²⁹ +x ²⁸ +x ²⁷ +x ²⁵ +x ²² +x ²⁰ +x ¹⁹ +x ¹⁴ +x ¹³ +x ¹² +x¹¹ +x ¹⁰ +x ⁸ +x ⁷ +x ⁶ +x ³ +x ⁰.

We assume a three level evaluation tree which is constructed inaccordance with the present invention as shown in FIG. 4. At the bottomlevel of the tree (the leaf nodes), we require the following powers ofr: r⁰, r¹, r² . . . r⁷. The odd powers are calculated r¹, r³, r⁵, and r⁷(by squaring r and 3 multiplications by r²).

When a above is decomposed in the tree, the leaf nodes are:L ₀=(r ⁷ +r ⁵ +r ³ +r ²+1)²L ₁ =r(r ⁷ +r ⁵ +r ³ +r ² +r)²L ₂=(r ⁷ +r ³)²L ₃ =r(r ⁶ +r ⁵ +r ⁴ +r ² +r+1)²To evaluate the leaf node L₀, we will evaluate the component polynomial,then square it and, when appropriate, further multiply its value by r toobtain the value of the leaf node:0) zero A1) add r¹ to A, square A, now A=r²2) add in r⁰, r³, r⁵, r⁷ to A3) square A=L₀For L₁, we will0) zero A1) add r¹ to A2) square A3) add r¹, r³, r⁵, r⁷, to A4) square A5) multiply A by r=Lfor L₂0) zero A1) add in r³, r⁷2) square A=L₂for L₃0) zero A1) add in r¹2) square A=r²3) add in r¹, r³4) square A=r⁶+r⁴+r²5) add in r⁰, r¹, r⁵

A=r⁶+r⁵+r⁴+r²+r+1

6) square A

7) multiply A by r=L₃

Now a(r) is built by evaluating the tree M₀=(L₀+L₁)², M₁=r(L₂+L₃)².Finally, a(r)=T₀=M₀+M₁.

Thus, it may be seen that his method may be applied to variouscryptographic schemes such as key exchange schemes, signature schemesand encryption schemes.

Although the invention has been described with reference to certainspecific embodiments, various modifications thereof will be apparent tothose skilled in the art without departing from the spirit and scope ofthe invention as outlined in the claims appended hereto. For example,the invention may be applied to basis conversion wherein the bases to beconverted between are an optimal normal basis and a polynomial basis.

1. (canceled)
 2. A computing device comprising an accumulator forevaluating a polynomial a(x) of a finite field of characteristic 2 at anelement r, said polynomial a(x) having a degree, said device beingconfigured to: a) initialize said accumulator to zero; b) obtain a valuer^(k) for each odd number k less than said degree; c) for each evennumber less than said degree and greater than 2 which is also anexponentiation of 2, if a corresponding coefficient is a one, add r¹ tosaid accumulator and square said accumulator; d) for all other evennumbers less than said degree, if the corresponding coefficient is aone, add a corresponding r^(k) value to said accumulator and square saidaccumulator; e) for each value r^(k) whose corresponding coefficient isa one, add the corresponding r^(k) value to said accumulator; f) if thecorresponding coefficient is a one, add d to said accumulator; and g)output a final value in said accumulator representing a(r).
 3. Acomputing device according to claim 2 wherein said values K arepre-computed and stored in a memory accessible by said computing device.4. A computing device according to claim 2 wherein said values P arecomputed by said device.
 5. A method for evaluating a polynomial a(x) ofa finite field of characteristic 2 at an element r, said polynomial a(x)having a degree, said method comprising the steps of: a) initializing anaccumulator to zero; b) obtaining a value r^(k) for each odd number kless than said degree; c) for each even number less than said degree andgreater than 2 which is also an exponentiation of 2, if a correspondingcoefficient is a one, adding r¹ to said accumulator and squaring saidaccumulator; d) for all other even numbers less than said degree, if thecorresponding coefficient is a one, adding a corresponding r^(k) valueto said accumulator and squaring said accumulator; e) for each valuer^(k) whose corresponding coefficient is a one, adding the correspondingr^(k) value to said accumulator; f) if the corresponding coefficient isa one, adding r⁰ to said accumulator; and g) providing a final valuefrom said accumulator to a cryptographic scheme representing a(r).
 6. Amethod according to claim 5 wherein said values r^(k) are pre-computedand stored in a memory accessible by a computing device.
 7. A methodaccording to claim 5 wherein said values r^(k) are computed by acomputing device during execution of said method.
 8. A computer readablemedium comprising computer executable instructions for performing themethod of claim
 5. 9. A computer device comprising an accumulator forevaluating a polynomial a(x) of a finite field of characteristic 3 at anelement r, said polynomial a(x) having a degree, said device beingconfigured to: a) initialize said accumulator to zero; b) for eachnumber i which is less than said degree and is an exponentiation of 3,in descending order, add a corresponding coefficient a_(i) multiplied byr¹ to said accumulator; c) cube said accumulator; d) obtain a valuer^(k) for each number k≠i which when multiplied by 3 is less than saiddegree; e) add each r^(k) multiplied by a corresponding coefficienta_(3k) to said accumulator; f) cube said accumulator; g) for all othercoefficients t less said degree, add each a^(t)r^(t) to saidaccumulator; and h) output a final value in said accumulatorrepresenting a(r).
 10. A computing device according to claim 9 whereinsaid values r^(k) are pre-computed and stored in a memory accessible bysaid computing device.
 11. A computing device according to claim 9wherein said values r^(k) are computed by said device.
 12. A method forevaluating a polynomial a(x) of a finite field of characteristic 3 at anelement r, said polynomial a(x) having a degree, said method comprisingthe steps of: a) initializing an accumulator to zero; b) for each numberi which is less than said degree and is an exponentiation of 3, indescending order, adding a corresponding coefficient a_(i) multiplied byr^(i) to said accumulator; c) cubing said accumulator; d) obtaining avalue r^(k) for each number k≠i which when multiplied by 3 is less thansaid degree; e) adding each r^(k) multiplied by a correspondingcoefficient a_(3k) to said accumulator; f) cubing said accumulator; g)for all other coefficients t less said degree, adding each a^(t)r^(t) tosaid accumulator; and h) outputting a final value in said accumulatorrepresenting a(r).
 13. A method according to claim 12 wherein saidvalues r^(k) are pre-computed and stored in a memory accessible by acomputing device.
 14. A method according to claim 12 wherein said valuesr^(k) are computed by a computing device during execution of saidmethod.
 15. A computer readable medium comprising computer executableinstructions for performing the method of claim
 12. 16. A method forevaluating a polynomial a(x) of a finite field of characteristic q at anelement r, said polynomial a(x) having a degree, said method comprisingthe steps of constructing a table of values using powers of q; andevaluating said polynomial a(x) by performing accumulation and qpowering steps until all required powers in said polynomial a(x) areadded in according to q and the degree of said polynomial a(x).